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chapter 8: estimating with confidence
8.1 confidence intervals: the basics
I'm thinking of a number...
Activity p468 ­­
I have selected an integer, M. Rather than guess this integer, you will guess an interval of numbers that might include my integer.
The command "mean(randNorm(M, 20, 16))" selects a sample of 16 integers from the Normal distribution, N(M, 20). Based on this one sample mean, what is a good range that includes M? How confident are you of your interval?
Point estimate: single best guess for the value of a population parameter, i.e., the results of a survey or experiment, p­hat or x­bar
It is VERY UNLIKELY that our estimate is correct!! So, let's find a range of values that might include it.
Back to the activity, M = _____. If I keep taking samples of size 16, how often will my sample fall within a good range? What would be a good range? How confident can we be at different ranges? Let's investigate...
Confidence Interval:
estimate ± margin of error
Margin of error: how close the estimate tends to be to the true
parameter in repeated sampling
Confidence level, C: percent of time that our (unbiased) sampling
method would result in an interval that captures the true
The confidence level does
NOT tell us the chance
that a particular
confidence interval
captures the population
Instead, the confidence interval gives us a
set of plausible values for the parameter.
CYU pg476
How much does the fat content of Brand X hot dogs vary? To find out, researchers measured the fat content (in grams) of a random sample of 10 Brand X hot dogs. A 95% confidence interval for the population standard deviation σ is 2.84 to 7.55.
1. Interpret the confidence interval.
1. We are 95% confident that the interval from 2.84 to 7.55 g captures the true standard deviation of the fat content of Brand X hot dogs.
2. Interpret the confidence level.
2. In 95% of all possible samples of 10 Brand X hot dogs, the resulting confidence interval would capture the true standard deviation.
3. True or false: The interval from 2.84 to 7.55 has a 95% chance of containing the actual population standard deviation σ. 3. False. The probability is either 1 (if the interval contains the true standard deviation) or 0 (if it doesn't).
Margin of Error = What this looks like for....
Sample means:
Sample proportions:
Consider the formulas...
How will the confidence interval (and margin of error) change as
n increases?
the confidence level (C) increases?
Gotta check conditions! We're assuming an approximately Normal
distribution, and we're sampling without replacing!